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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A characterization of metric completeness


Author: J. D. Weston
Journal: Proc. Amer. Math. Soc. 64 (1977), 186-188
MSC: Primary 54C30
MathSciNet review: 0458359
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Abstract: A proof is given of a theorem, relevant to fixed-point theory, which implies that a metric space (X, d) is complete if and only if, for each continuous function $ h:X \to {\mathbf{R}}$ bounded below on X, there is a point $ {x_0}$ such that $ h({x_0}) - h(x) < d({x_0},x)$ for every other point x.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0458359-2
PII: S 0002-9939(1977)0458359-2
Keywords: Completeness, fixed point, metric space, order, semicontinuity
Article copyright: © Copyright 1977 American Mathematical Society